Something that you will eventually get and continue to get once you become a Marine. Getting the "big green weenie" may include such fine events as: getting fucked with duty on a particularly important weekend, getting in trouble for some small petty shit while being in garrison, getting stationed at a shithole like 29 Dumps.
The "big" represents the big bad Marine Corps picking on the little guy, the Marine. The "green" represents the true color of the Corps. The "weenie" represents getting fucked over.

LCpl Hooligan: What I got duty again?! Dude Im getting fucked in the ass by the big green weenie!